Sine rule is an important rule relating to the sides and angles of any triangle. Here is a Subjective problem no. 120 from TOMATO. Try it.
Problem: Sine Rule and Triangle
(i) If , show that
(ii) Let triangles ABC and DEF be inscribed in the same circle. If the triangles are of equal perimeter, then prove that
(iii) State and prove the converse of (ii) above
(i) We know transformation formula from trigonometry
Now we know that
Since the two triangles are inscribed in the same circle, they must have the same circumradius. Let the common circumradius be R. If a, b, c, d, e, f be the sides opposite to the sides BC, CA, AB, EF, DF, DE respectively, then using the rule of sines we can say,
As the perimeter of the triangle are equal, hence a+b+c = d+e+f. This implies
We apply the sine rule in reverse order to get the converse.